Rational function

Definition

A rational function is a function that can be expressed as the quotient of two polynomials, where the denominator is not zero. It has the form $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials.

5 Must Know Facts For Your Next Test

1. The domain of a rational function excludes values that make the denominator zero.
2. Vertical asymptotes occur at values of $x$ that make the denominator zero (provided these values do not cancel out with the numerator).
3. Horizontal asymptotes depend on the degrees of the numerator and denominator polynomials.
4. A hole in the graph occurs at any value that cancels out in both the numerator and denominator.
5. Rational functions can exhibit end behavior similar to polynomial functions based on their leading terms.

Review Questions

• What is a vertical asymptote and how do you find it for a rational function?
• How do you determine if there is a hole in the graph of a rational function?
• What conditions determine whether a rational function has a horizontal asymptote?

Related terms

Polynomial Function:

A function that can be expressed in the form $P(x) = a_n x^n + ... + a_1 x + a_0$, where $a_i$ are constants and $n$ is a non-negative integer.

Asymptote:

A line that a graph approaches but never touches or crosses. Types include vertical, horizontal, and oblique.

Domain:

The set of all possible input values (x-values) for which the function is defined.